将多项式曲线转换为贝塞尔曲线控制点

问题描述

如何以幂形式计算给定曲线的控制点?假设我有 p(t)=(x(t),y(t)) 和 4 个控制点.

x(t) = 2ty(t) = (t^3)+3(t^2)

解决方案

您始终可以从幂基转换为 Bernstein 基.这总是可行的，并且会给你准确的结果.请参阅此链接的第 3.3 节(

其中 M 是 Berstein 基的阶数，0

以常见的三次 Berstein 基(其中 M=3)为例，我们将有

How do I compute the control points given a curve in the form of power form? Say I have p(t)=(x(t),y(t)) and 4 control points.

x(t) = 2t
y(t) = (t^3)+3(t^2)

解决方案

You can always convert from power basis to Bernstein basis. This is always doable and will give you the precise result. Refer to section 3.3 of this link (http://cagd.cs.byu.edu/~557/text/ch3.pdf) for details.

EDIT:Since the above link is no longer available, I am listing the formula below:

where M is the degree of the Berstein basis, 0 <= k <= M and b_i,k=0 if i < k.

Using the common cubic Berstein basis (where M=3) as an example, we will have

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